IODS course project
Valeria Caras
About the project_exercise1
Write a short description about the course and add a link to your GitHub repository here. This is an R Markdown (.Rmd) file so you can use R Markdown syntax.
The course Introduction to Open Data Science
This course intrigues me and I expect it to be very useful for my following research. The content of the course is very interesting that is why I have registered for the course. The course has very friendly and open environment.
Exercise 2
Describe the work you have done this week and summarize your learning.
- Describe your work and results clearly.
- Assume the reader has an introductory course level understanding of writing and reading R code as well as statistical methods.
- Assume the reader has no previous knowledge of your data or the more advanced methods you are using.
1.This is the data from the international survey of Approaches to Learning. It icludes the attitute toward statistics, exam points, age, gender and anwers to the guestions which were combined in the theree variables as deep,surface and strategic questions. Also the observations where the exam points variable is zero were excluded.The data included 166 observations and 7 variables.
The plot illustrates the normality of the errors disributions.The normal distributions implies that the majority of the obserbations are located close to the regression line. Here only a part of the observations are close what is why normality is questionable. The plot illustrates the relationships between the variables.For example, we can see a negative correlation between points and age, surf and stra, points and surf and etc. There are strong correlations between points and age, age and attitude, age and deep, attitude and deep, age and stra, attitude and stra, deep and stra and etc.
I have created a model with the dependent variable points and independent variables attitude, age and stra. The attitude variable has a high significance (p-value).It has 3.480 estimated coefficient with the positive relations with the dependent variable (when the value of the independent variable rises dependent also rises). Age and stra veriables are significant on 0.05 level. Age has a negative etsimate that means that when it rises the points fall. Srta has a positive relation with the points.
The R-square is not very big since the model predicts around 20% of the points.F statistics compares the model with the model with only intercept. Here it has significant p-value which means that my model predicts better than a model with no predictors.
5.Residuals vs Fitted values plot checks the assumption that size on errors should not depend on the explanotary variables.Here only a small part of the obseravations do not fit the pattern that is why there is no problem with the assumption. Normal QQ-plot shows that the erros of the model are normally distributed since points mostly fall on regression line. Residuals vs Leverage diahnostic measures the impact of the single observation to the model. In the plot we can see several observations with the higher leverage which are located too below the line (-4-2) when the line is around 0.
chooseCRANmirror(graphics=FALSE, ind=1)
install.packages("lmtest", repos = "http://cran.us.r-project.org")##
## The downloaded binary packages are in
<<<<<<< HEAD
## /var/folders/8t/x1h4shkx7ms9f1___g0qnkhr0000gn/T//Rtmpa7Z52d/downloaded_packageslibrary(lmtest)## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numericlearning2014 <- read.table("http://s3.amazonaws.com/assets.datacamp.com/production/course_2218/datasets/learning2014.txt", sep=",", header=TRUE)
learning2014## gender age attitude deep stra surf points
## 1 F 53 3.7 3.583333 3.375 2.583333 25
## 2 M 55 3.1 2.916667 2.750 3.166667 12
## 3 F 49 2.5 3.500000 3.625 2.250000 24
## 4 M 53 3.5 3.500000 3.125 2.250000 10
## 5 M 49 3.7 3.666667 3.625 2.833333 22
## 6 F 38 3.8 4.750000 3.625 2.416667 21
## 7 M 50 3.5 3.833333 2.250 1.916667 21
## 8 F 37 2.9 3.250000 4.000 2.833333 31
## 9 M 37 3.8 4.333333 4.250 2.166667 24
## 10 F 42 2.1 4.000000 3.500 3.000000 26
## 11 M 37 3.9 3.583333 3.625 2.666667 31
## 12 F 34 3.8 3.833333 4.750 2.416667 31
## 13 F 34 2.4 4.250000 3.625 2.250000 23
## 14 F 34 3.0 3.333333 3.500 2.750000 25
## 15 M 35 2.6 4.166667 1.750 2.333333 21
## 16 F 33 4.1 3.666667 3.875 2.333333 31
## 17 F 32 2.6 4.083333 1.375 2.916667 20
## 18 F 44 2.6 3.500000 3.250 2.500000 22
## 19 M 29 1.7 4.083333 3.000 3.750000 9
## 20 F 30 2.7 4.000000 3.750 2.750000 24
## 21 M 27 3.9 3.916667 2.625 2.333333 28
## 22 M 29 3.4 4.000000 2.375 2.416667 30
## 23 F 31 2.7 4.000000 3.625 3.000000 24
## 24 F 37 2.3 3.666667 2.750 2.416667 9
## 25 F 26 3.7 3.666667 1.750 2.833333 26
## 26 F 26 4.4 4.416667 3.250 3.166667 32
## 27 M 30 4.1 3.916667 4.000 3.000000 32
## 28 F 33 3.7 3.750000 3.625 2.000000 33
## 29 F 33 2.5 3.250000 2.875 3.500000 29
## 30 M 28 3.0 3.583333 3.000 3.750000 30
## 31 M 26 3.4 4.916667 1.625 2.500000 19
## 32 F 27 3.2 3.583333 3.250 2.083333 23
## 33 F 25 2.0 2.916667 3.500 2.416667 19
## 34 F 31 2.4 3.666667 3.000 2.583333 12
## 35 M 20 4.2 4.500000 3.250 1.583333 10
## 36 F 39 1.6 4.083333 1.875 2.833333 11
## 37 M 38 3.1 3.833333 4.375 1.833333 20
## 38 M 24 3.8 3.250000 3.625 2.416667 26
## 39 M 26 3.8 2.333333 2.500 3.250000 31
## 40 M 25 3.3 3.333333 1.250 3.416667 20
## 41 F 30 1.7 4.083333 4.000 3.416667 23
## 42 F 25 2.5 2.916667 3.000 3.166667 12
## 43 M 30 3.2 3.333333 2.500 3.500000 24
## 44 F 48 3.5 3.833333 4.875 2.666667 17
## 45 F 24 3.2 3.666667 5.000 2.416667 29
## 46 F 40 4.2 4.666667 4.375 3.583333 23
## 47 M 25 3.1 3.750000 3.250 2.083333 28
## 48 F 23 3.9 3.416667 4.000 3.750000 31
## 49 F 25 1.9 4.166667 3.125 2.916667 23
## 50 F 23 2.1 2.916667 2.500 2.916667 25
## 51 M 27 2.5 4.166667 3.125 2.416667 18
## 52 M 25 3.2 3.583333 3.250 3.000000 19
## 53 M 23 3.2 2.833333 2.125 3.416667 22
## 54 F 23 2.6 4.000000 2.750 2.916667 25
## 55 F 23 2.3 2.916667 2.375 3.250000 21
## 56 F 45 3.8 3.000000 3.125 3.250000 9
## 57 F 22 2.8 4.083333 4.000 2.333333 28
## 58 F 23 3.3 2.916667 4.000 3.250000 25
## 59 M 21 4.8 3.500000 2.250 2.500000 29
## 60 M 21 4.0 4.333333 3.250 1.750000 33
## 61 F 21 4.0 4.250000 3.625 2.250000 33
## 62 F 21 4.7 3.416667 3.625 2.083333 25
## 63 F 26 2.3 3.083333 2.500 2.833333 18
## 64 F 25 3.1 4.583333 1.875 2.833333 22
## 65 F 26 2.7 3.416667 2.000 2.416667 17
## 66 M 21 4.1 3.416667 1.875 2.250000 25
## 67 F 23 3.4 3.416667 4.000 2.833333 28
## 68 F 22 2.5 3.583333 2.875 2.250000 22
## 69 F 22 2.1 1.583333 3.875 1.833333 26
## 70 F 22 1.4 3.333333 2.500 2.916667 11
## 71 F 23 1.9 4.333333 2.750 2.916667 29
## 72 M 22 3.7 4.416667 4.500 2.083333 22
## 73 M 23 3.2 4.833333 3.375 2.333333 21
## 74 M 24 2.8 3.083333 2.625 2.416667 28
## 75 F 22 4.1 3.000000 4.125 2.750000 33
## 76 F 23 2.5 4.083333 2.625 3.250000 16
## 77 M 22 2.8 4.083333 2.250 1.750000 31
## 78 M 20 3.8 3.750000 2.750 2.583333 22
## 79 M 22 3.1 3.083333 3.000 3.333333 31
## 80 M 21 3.5 4.750000 1.625 2.833333 23
## 81 F 22 3.6 4.250000 1.875 2.500000 26
## 82 F 23 2.6 4.166667 3.375 2.416667 12
## 83 M 21 4.4 4.416667 3.750 2.416667 26
## 84 M 22 4.5 3.833333 2.125 2.583333 31
## 85 M 29 3.2 3.333333 2.375 3.000000 19
## 86 F 29 3.9 3.166667 2.750 2.000000 30
## 87 F 21 2.5 3.166667 3.125 3.416667 12
## 88 M 28 3.3 3.833333 3.500 2.833333 17
## 89 F 21 3.3 4.250000 2.625 2.250000 18
## 90 F 30 3.0 3.833333 3.375 2.750000 19
## 91 F 21 2.9 3.666667 2.250 3.916667 21
## 92 M 23 3.3 3.833333 3.000 2.333333 24
## 93 F 21 3.3 3.833333 4.000 2.750000 28
## 94 F 21 3.5 3.833333 3.500 2.750000 17
## 95 F 20 3.6 3.666667 2.625 2.916667 18
## 96 M 22 3.7 4.333333 2.500 2.083333 17
## 97 M 21 4.2 3.750000 3.750 3.666667 23
## 98 M 21 3.2 4.166667 3.625 2.833333 26
## 99 F 20 5.0 4.000000 4.125 3.416667 28
## 100 M 22 4.7 4.000000 4.375 1.583333 31
## 101 F 20 3.6 4.583333 2.625 2.916667 27
## 102 F 20 3.6 3.666667 4.000 3.000000 25
## 103 M 24 2.9 3.666667 2.750 2.916667 23
## 104 F 20 3.5 3.833333 2.750 2.666667 21
## 105 F 19 4.0 2.583333 1.375 3.000000 27
## 106 F 21 3.5 3.500000 2.250 2.750000 28
## 107 F 21 3.2 3.083333 3.625 3.083333 23
## 108 F 22 2.6 4.250000 3.750 2.500000 21
## 109 F 25 2.0 3.166667 4.000 2.333333 25
## 110 F 21 2.7 3.083333 3.125 3.000000 11
## 111 F 22 3.2 4.166667 3.250 3.000000 19
## 112 F 25 3.3 2.250000 2.125 4.000000 24
## 113 F 20 3.9 3.333333 2.875 3.250000 28
## 114 M 24 3.3 3.083333 1.500 3.500000 21
## 115 F 20 3.0 2.750000 2.500 3.500000 24
## 116 M 21 3.7 3.250000 3.250 3.833333 24
## 117 F 20 2.5 4.000000 3.625 2.916667 20
## 118 F 20 2.9 3.583333 3.875 2.166667 19
## 119 M 31 3.9 4.083333 3.875 1.666667 30
## 120 F 20 3.6 4.250000 2.375 2.083333 22
## 121 F 22 2.9 3.416667 3.000 2.833333 16
## 122 F 22 2.1 3.083333 3.375 3.416667 16
## 123 M 21 3.1 3.500000 2.750 3.333333 19
## 124 M 22 4.0 3.666667 4.500 2.583333 30
## 125 F 21 3.1 4.250000 2.625 2.833333 23
## 126 F 21 2.3 4.250000 2.750 3.333333 19
## 127 F 21 2.8 3.833333 3.250 3.000000 18
## 128 F 21 3.7 4.416667 4.125 2.583333 28
## 129 F 20 2.6 3.500000 3.375 2.416667 21
## 130 F 21 2.4 3.583333 2.750 3.583333 19
## 131 F 25 3.0 3.666667 4.125 2.083333 27
## 132 M 21 2.8 2.083333 3.250 4.333333 24
## 133 F 24 2.9 4.250000 2.875 2.666667 21
## 134 F 20 2.4 3.583333 2.875 3.000000 20
## 135 M 21 3.1 4.000000 2.375 2.666667 28
## 136 F 20 1.9 3.333333 3.875 2.166667 12
## 137 F 20 2.0 3.500000 2.125 2.666667 21
## 138 F 18 3.8 3.166667 4.000 2.250000 28
## 139 F 21 3.4 3.583333 3.250 2.666667 31
## 140 F 19 3.7 3.416667 2.625 3.333333 18
## 141 F 21 2.9 4.250000 2.750 3.500000 25
## 142 F 20 2.3 3.250000 4.000 2.750000 19
## 143 M 21 4.1 4.416667 3.000 2.000000 21
## 144 F 20 2.7 3.250000 3.375 2.833333 16
## 145 F 21 3.5 3.916667 3.875 3.500000 7
## 146 F 20 3.4 3.583333 3.250 2.500000 21
## 147 F 18 3.2 4.500000 3.375 3.166667 17
## 148 M 22 3.3 3.583333 4.125 3.083333 22
## 149 F 22 3.3 3.666667 3.500 2.916667 18
## 150 M 24 3.5 2.583333 2.000 3.166667 25
## 151 F 19 3.2 4.166667 3.625 2.500000 24
## 152 F 20 3.1 3.250000 3.375 3.833333 23
## 153 F 20 2.8 4.333333 2.125 2.250000 23
## 154 F 17 1.7 3.916667 4.625 3.416667 26
## 155 M 19 1.9 2.666667 2.500 3.750000 12
## 156 F 20 3.5 3.083333 2.875 3.000000 32
## 157 F 20 2.4 3.750000 2.750 2.583333 22
## 158 F 20 2.1 4.166667 4.000 3.333333 20
## 159 F 20 2.9 4.166667 2.375 2.833333 21
## 160 F 19 1.9 3.250000 3.875 3.000000 23
## 161 F 19 2.0 4.083333 3.375 2.833333 20
## 162 F 22 4.2 2.916667 1.750 3.166667 28
## 163 M 35 4.1 3.833333 3.000 2.750000 31
## 164 F 18 3.7 3.166667 2.625 3.416667 18
## 165 F 19 3.6 3.416667 2.625 3.000000 30
## 166 M 21 1.8 4.083333 3.375 2.666667 19str(learning2014)## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: num 3.7 3.1 2.5 3.5 3.7 3.8 3.5 2.9 3.8 2.1 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...dim(learning2014)## [1] 166 7library(ggplot2)
#initialize plot with data and aesthetic mapping
p1 <- ggplot(learning2014, aes(x = attitude, y = points))
#define the visualization type (points)
p2 <- p1 + geom_point()
#add a regression line
p3 <- p2 + geom_smooth(method = "lm")
#add a main title and draw the plot
p4 <- p3 + ggtitle ("Student's attitude versus exam points")
p4
#draw a scatter plot matrix of the variables in learning2014.
#[-1] excludes the first column (gender)
pairs(learning2014[-1])
#access the GGally and ggplot2 libraries
install.packages("GGally")##
## The downloaded binary packages are in
<<<<<<< HEAD
## /var/folders/8t/x1h4shkx7ms9f1___g0qnkhr0000gn/T//Rtmpa7Z52d/downloaded_packagesinstall.packages("ggplot2")##
## The downloaded binary packages are in
## /var/folders/8t/x1h4shkx7ms9f1___g0qnkhr0000gn/T//Rtmpa7Z52d/downloaded_packagesinstall.packages("ggplot2")##
## The downloaded binary packages are in
## /var/folders/8t/x1h4shkx7ms9f1___g0qnkhr0000gn/T//RtmpewQAQu/downloaded_packageslibrary(GGally)## Registered S3 method overwritten by 'GGally':
## method from
## +.gg ggplot2library(ggplot2)
# create a more advanced plot matrix with ggpairs()
p <- ggpairs(learning2014, mapping = aes(), lower = list(combo = wrap("facethist", bins = 20)))
p
# fit a linear model
my_model <- lm(points ~ 1, data = learning2014)
library(ggplot2)
qplot(attitude, points, data = learning2014) + geom_smooth(method = "lm")
qplot(attitude, age, data = learning2014) + geom_smooth(method = "lm")
qplot(attitude, stra, data = learning2014) + geom_smooth(method = "lm")
# create an plot matrix with ggpairs()
ggpairs(learning2014, lower = list(combo = wrap("facethist", bins = 20)))
#create a regression model with 3 multiple explanatory variables
my_model1 <- lm(points ~ attitude + age + stra, data = learning2014)
summary(my_model1)##
## Call:
## lm(formula = points ~ attitude + age + stra, data = learning2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -18.1149 -3.2003 0.3303 3.4129 10.7599
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 10.89543 2.64834 4.114 6.17e-05 ***
## attitude 3.48077 0.56220 6.191 4.72e-09 ***
## age -0.08822 0.05302 -1.664 0.0981 .
## stra 1.00371 0.53434 1.878 0.0621 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.26 on 162 degrees of freedom
## Multiple R-squared: 0.2182, Adjusted R-squared: 0.2037
## F-statistic: 15.07 on 3 and 162 DF, p-value: 1.07e-08par(mfrow = c(2,2))
plot(my_model1)
Exercise 3
alc <- read.csv( file="data/alc.csv", sep=",")
dim(alc)## [1] 382 35str(alc)## 'data.frame': 382 obs. of 35 variables:
## $ school : Factor w/ 2 levels "GP","MS": 1 1 1 1 1 1 1 1 1 1 ...
## $ sex : Factor w/ 2 levels "F","M": 1 1 1 1 1 2 2 1 2 2 ...
## $ age : int 18 17 15 15 16 16 16 17 15 15 ...
## $ address : Factor w/ 2 levels "R","U": 2 2 2 2 2 2 2 2 2 2 ...
## $ famsize : Factor w/ 2 levels "GT3","LE3": 1 1 2 1 1 2 2 1 2 1 ...
## $ Pstatus : Factor w/ 2 levels "A","T": 1 2 2 2 2 2 2 1 1 2 ...
## $ Medu : int 4 1 1 4 3 4 2 4 3 3 ...
## $ Fedu : int 4 1 1 2 3 3 2 4 2 4 ...
## $ Mjob : Factor w/ 5 levels "at_home","health",..: 1 1 1 2 3 4 3 3 4 3 ...
## $ Fjob : Factor w/ 5 levels "at_home","health",..: 5 3 3 4 3 3 3 5 3 3 ...
## $ reason : Factor w/ 4 levels "course","home",..: 1 1 3 2 2 4 2 2 2 2 ...
## $ nursery : Factor w/ 2 levels "no","yes": 2 1 2 2 2 2 2 2 2 2 ...
## $ internet : Factor w/ 2 levels "no","yes": 1 2 2 2 1 2 2 1 2 2 ...
## $ guardian : Factor w/ 3 levels "father","mother",..: 2 1 2 2 1 2 2 2 2 2 ...
## $ traveltime: int 2 1 1 1 1 1 1 2 1 1 ...
## $ studytime : int 2 2 2 3 2 2 2 2 2 2 ...
## $ failures : int 0 0 2 0 0 0 0 0 0 0 ...
## $ schoolsup : Factor w/ 2 levels "no","yes": 2 1 2 1 1 1 1 2 1 1 ...
## $ famsup : Factor w/ 2 levels "no","yes": 1 2 1 2 2 2 1 2 2 2 ...
## $ paid : Factor w/ 2 levels "no","yes": 1 1 2 2 2 2 1 1 2 2 ...
## $ activities: Factor w/ 2 levels "no","yes": 1 1 1 2 1 2 1 1 1 2 ...
## $ higher : Factor w/ 2 levels "no","yes": 2 2 2 2 2 2 2 2 2 2 ...
## $ romantic : Factor w/ 2 levels "no","yes": 1 1 1 2 1 1 1 1 1 1 ...
## $ famrel : int 4 5 4 3 4 5 4 4 4 5 ...
## $ freetime : int 3 3 3 2 3 4 4 1 2 5 ...
## $ goout : int 4 3 2 2 2 2 4 4 2 1 ...
## $ Dalc : int 1 1 2 1 1 1 1 1 1 1 ...
## $ Walc : int 1 1 3 1 2 2 1 1 1 1 ...
## $ health : int 3 3 3 5 5 5 3 1 1 5 ...
## $ absences : int 5 3 8 1 2 8 0 4 0 0 ...
## $ G1 : int 2 7 10 14 8 14 12 8 16 13 ...
## $ G2 : int 8 8 10 14 12 14 12 9 17 14 ...
## $ G3 : int 8 8 11 14 12 14 12 10 18 14 ...
## $ alc_use : num 1 1 2.5 1 1.5 1.5 1 1 1 1 ...
## $ high_use : logi FALSE FALSE TRUE FALSE FALSE FALSE ...This is a combined data from two original datasets on student achievement in secondary education of two Portuguese schools. It was merged to include only the students who answered the questionnaire in both math and portuguese classes.New column high use was created when’alc_use’ is greater than 2. Data has 382 observations of 35 variables. I choose the 4 following variables to test their relationships with high alcohol consumption. 1. Pstatus - parent’s cohabitation status (binary: ‘T’ - living together or ‘A’ - apart. I suppose that living apart with the students leads to high alcohol consumption.
famrel - quality of family relationships (numeric: from 1 - very bad to 5 - excellent). I suppose that bad family relationships result on high alcohol consumption.
goout - going out with friends (numeric: from 1 - very low to 5 - very high). I suppose that student who often go out with friends consume more alcohol.
romantic - with a romantic relationship (binary: yes or no). I suppose that students who are not in relationships consume more alcohol.
library(tidyr); library(dplyr); library(ggplot2)##
## Attaching package: 'dplyr'## The following object is masked from 'package:GGally':
##
## nasa## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionglimpse(alc)## Observations: 382
## Variables: 35
## $ school <fct> GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, GP, G…
## $ sex <fct> F, F, F, F, F, M, M, F, M, M, F, F, M, M, M, F, F, F,…
## $ age <int> 18, 17, 15, 15, 16, 16, 16, 17, 15, 15, 15, 15, 15, 1…
## $ address <fct> U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U, U,…
## $ famsize <fct> GT3, GT3, LE3, GT3, GT3, LE3, LE3, GT3, LE3, GT3, GT3…
## $ Pstatus <fct> A, T, T, T, T, T, T, A, A, T, T, T, T, T, A, T, T, T,…
## $ Medu <int> 4, 1, 1, 4, 3, 4, 2, 4, 3, 3, 4, 2, 4, 4, 2, 4, 4, 3,…
## $ Fedu <int> 4, 1, 1, 2, 3, 3, 2, 4, 2, 4, 4, 1, 4, 3, 2, 4, 4, 3,…
## $ Mjob <fct> at_home, at_home, at_home, health, other, services, o…
## $ Fjob <fct> teacher, other, other, services, other, other, other,…
## $ reason <fct> course, course, other, home, home, reputation, home, …
## $ nursery <fct> yes, no, yes, yes, yes, yes, yes, yes, yes, yes, yes,…
## $ internet <fct> no, yes, yes, yes, no, yes, yes, no, yes, yes, yes, y…
## $ guardian <fct> mother, father, mother, mother, father, mother, mothe…
## $ traveltime <int> 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 2, 1, 1, 1, 3,…
## $ studytime <int> 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 3, 1, 3, 2,…
## $ failures <int> 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,…
## $ schoolsup <fct> yes, no, yes, no, no, no, no, yes, no, no, no, no, no…
## $ famsup <fct> no, yes, no, yes, yes, yes, no, yes, yes, yes, yes, y…
## $ paid <fct> no, no, yes, yes, yes, yes, no, no, yes, yes, yes, no…
## $ activities <fct> no, no, no, yes, no, yes, no, no, no, yes, no, yes, y…
## $ higher <fct> yes, yes, yes, yes, yes, yes, yes, yes, yes, yes, yes…
## $ romantic <fct> no, no, no, yes, no, no, no, no, no, no, no, no, no, …
## $ famrel <int> 4, 5, 4, 3, 4, 5, 4, 4, 4, 5, 3, 5, 4, 5, 4, 4, 3, 5,…
## $ freetime <int> 3, 3, 3, 2, 3, 4, 4, 1, 2, 5, 3, 2, 3, 4, 5, 4, 2, 3,…
## $ goout <int> 4, 3, 2, 2, 2, 2, 4, 4, 2, 1, 3, 2, 3, 3, 2, 4, 3, 2,…
## $ Dalc <int> 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,…
## $ Walc <int> 1, 1, 3, 1, 2, 2, 1, 1, 1, 1, 2, 1, 3, 2, 1, 2, 2, 1,…
## $ health <int> 3, 3, 3, 5, 5, 5, 3, 1, 1, 5, 2, 4, 5, 3, 3, 2, 2, 4,…
## $ absences <int> 5, 3, 8, 1, 2, 8, 0, 4, 0, 0, 1, 2, 1, 1, 0, 5, 8, 3,…
## $ G1 <int> 2, 7, 10, 14, 8, 14, 12, 8, 16, 13, 12, 10, 13, 11, 1…
## $ G2 <int> 8, 8, 10, 14, 12, 14, 12, 9, 17, 14, 11, 12, 14, 11, …
## $ G3 <int> 8, 8, 11, 14, 12, 14, 12, 10, 18, 14, 12, 12, 13, 12,…
## $ alc_use <dbl> 1.0, 1.0, 2.5, 1.0, 1.5, 1.5, 1.0, 1.0, 1.0, 1.0, 1.5…
## $ high_use <lgl> FALSE, FALSE, TRUE, FALSE, FALSE, FALSE, FALSE, FALSE…#use gather() to gather columns into key-value pairs and then glimpse() at the resulting data
gather(alc) %>% glimpse## Warning: attributes are not identical across measure variables;
## they will be dropped## Observations: 13,370
## Variables: 2
## $ key <chr> "school", "school", "school", "school", "school", "school"…
## $ value <chr> "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP", "GP"…#draw a bar plot of each variable
gather(alc) %>% ggplot(aes(value)) + facet_wrap("key", scales = "free")+geom_bar()## Warning: attributes are not identical across measure variables;
## they will be dropped
Bar plots illustrate the following distribution of the variables: Pstatus: the majotiry of students have parents living together romantic: twice more students are no in romantic relations famrel: the majority of the respondents have very good relations with parents,less have excellent and good relations while thre are students who have bad and very bad relations goout: has a normal distribution.
library(dplyr); library(ggplot2)
# produce summary statistics by group
alc %>% group_by(sex, high_use) %>% summarise(count = mean(G3))## # A tibble: 4 x 3
## # Groups: sex [2]
## sex high_use count
## <fct> <lgl> <dbl>
## 1 F FALSE 11.4
## 2 F TRUE 11.7
## 3 M FALSE 12.2
## 4 M TRUE 10.3library(ggplot2)
#initialize a plot of high_use and G3
g1 <- ggplot(alc, aes(x = high_use, y = G3))
#define the plot as a boxplot and draw it
g1 + geom_boxplot() + ylab("grade")
#initialise a plot of high_use and absences
g2 <- ggplot(alc, aes(x = high_use, y = absences))
#define the plot as a boxplot and draw it
g2 + geom_boxplot() + ylab("Student absences by alcohol consumption and sex")
#my boxplots
g3 <- ggplot(alc, aes(x = high_use, y = goout))
g3 + geom_boxplot() + ylab("Go out by alcohol consumption")
g4 <- ggplot(alc, aes(x = high_use, y = famrel))
g4 + geom_boxplot() + ylab("Family relations by alcohol consumption")
First here is the boxplot from the exercise which shows the relations between high use of alcohol and final grades. Here we seen a lot of outliners with the low grades and several with high grades who high consume alcohol. Next there are boxplots on absences. Both boxpots are located quite low which shows the low level of abcences and true assumtion on higher alcohol use. However there are a lot of outliners who do not miss school and consume more as well as go to school and do not consume.
I have constructed the boxplot on goout and high use of alcohol. The true has a higher probability with the minimum 3 and max 5 which is often.The firts boxplot shows that 75% observanions are located low from 1 to 3 (students who go out rare and for them high use of alcohol is false). However there is one outliner which goes out often (5) and does not drink. -> I have support for my hypothesis about go out and high alcohol.
I have constructed the boxplot on family relations and high use of alcohol. In the fist plot we see that the asumption is false for people who have good (3) and excellent relations with parents (5) which coincides with my assumption + we have outliners who have bad relations with parents. The second boxplot illustrates that the assumtion is true for people who have bad and good relations with one outliner who has bad relations but does not consume much -> I have support for my hypothesis about family relations and high alcohol.
I also have tried to construct boxpolts for variables Pstatus and romantic but they look very stange. -> I suppose there are not significant relations between Pstatus and romantic relations with high alcohol consumption but let’s test it anyway in regression.
#find the model with glm()
m <- glm(high_use ~ famrel + goout+romantic+Pstatus, data = alc, family = "binomial")
#print out a summary of the model
summary(m)##
## Call:
## glm(formula = high_use ~ famrel + goout + romantic + Pstatus,
## family = "binomial", data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.5849 -0.7766 -0.5662 0.9554 2.4378
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.8376 0.7049 -2.607 0.00914 **
## famrel -0.3732 0.1340 -2.784 0.00537 **
## goout 0.7959 0.1185 6.718 1.84e-11 ***
## romanticyes -0.2826 0.2658 -1.063 0.28763
## PstatusT -0.1019 0.4029 -0.253 0.80027
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 406.92 on 377 degrees of freedom
## AIC: 416.92
##
## Number of Fisher Scoring iterations: 4# print out the coefficients of the model
coef(m)## (Intercept) famrel goout romanticyes PstatusT
## -1.8375812 -0.3731507 0.7959347 -0.2826057 -0.1019220#compute odds ratios (OR)
OR <- coef(m) %>% exp
#compute confidence intervals (CI)
CI <- confint(m)## Waiting for profiling to be done...#print out the odds ratios with their confidence intervals
cbind(OR, CI)## OR 2.5 % 97.5 %
## (Intercept) 0.1592020 -3.2514939 -0.4772475
## famrel 0.6885615 -0.6385795 -0.1114287
## goout 2.2165118 0.5699984 1.0354467
## romanticyes 0.7538169 -0.8126779 0.2317825
## PstatusT 0.9031000 -0.8753200 0.7157342In my model we see that only famrel and goout variables are significant. Famrel coeffcient is -0.3731507 which according to the scale of the variable means that better family relations lead to less alcohol use. Goout coefficient is 0.7959347 which means that often go out with friens leads to higher alcohol consumotion. So my hypothesis regarding these variables are testified while regarding status of parents and romantic relations not. <<<<<<< HEAD Odds show that go out twice increases the probability of alcohol consumption. Good family relations decrease the alcohol consumption by 68%, romantic relations also decrease the alcohol by 75% as well as higher status pf parents by 90%. ======= Odds show a higher than 1 probability of success (high alcohol consumption) for goout. Odds for presence of famreal is also higher (97.5% interval) than absence. >>>>>>> 1ec0b112ee3f4f518ef03287b10fa255ad0529af
#Binary predictions
# predict() the probability of high_use
probabilities <- predict(m, type = "response")
#add the predicted probabilities to 'alc'
alc <- mutate(alc, probability = probabilities)
#use the probabilities to make a prediction of high_use
alc <- mutate(alc, prediction = "high_use")
#see the last ten original classes, predicted probabilities, and class predictions
select(alc, famrel, goout, high_use, probability, prediction) %>% tail(10)## famrel goout high_use probability prediction
## 373 4 2 FALSE 0.13702332 high_use
## 374 5 3 TRUE 0.19506126 high_use
## 375 5 3 FALSE 0.19506126 high_use
## 376 4 3 FALSE 0.20967135 high_use
## 377 5 2 FALSE 0.09855474 high_use
## 378 4 4 FALSE 0.43822544 high_use
## 379 2 2 FALSE 0.25087798 high_use
## 380 1 1 FALSE 0.17994514 high_use
## 381 2 5 TRUE 0.78480132 high_use
## 382 4 1 TRUE 0.06684647 high_use#tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$high_use)## prediction
## high_use FALSE TRUE
## FALSE 268 0
## TRUE 0 114# access dplyr and ggplot2
library(dplyr); library(ggplot2)
# initialize a plot of 'high_use' versus 'probability' in 'alc'
g6 <- ggplot(alc, aes(x = "high_use", y = "probability"))
# define the geom as points and draw the plot
g6+geom_point()
# tabulate the target variable versus the predictions
table(high_use = alc$high_use, prediction = alc$prediction)%>%prop.table%>%addmargins## prediction
## high_use high_use Sum
## FALSE 0.7015707 0.7015707
## TRUE 0.2984293 0.2984293
## Sum 1.0000000 1.0000000#accuracy and loss function
# define a loss function (mean prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
# call loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = alc$high_use, prob = 0)## [1] 0.2984293loss_func(class = alc$high_use, prob = 1)## [1] 0.7015707loss_func(class = alc$high_use, prob = alc$probability)## [1] 0.2513089I construct a ‘confusion mattrix’ which describes the performance of my model with such predictors as famrel and goout. We see that the majority of cases (268) are cases in which we predicted False and are actually False. There are no cases (0) where we predicted True, but actual is False (so no Type I error) and no cases where we predicted False, True in actual (no Type II error). And there are 114 cases where we predicted True, and True in actual. So we can conlude that my model well predicts the target variable. Next I computed the total proportion of inaccurately classified individuals. We see that the average number of wrong predictions is 0.2984, while the correct predictions is 0.7015707. So the model predicts quite good.
#Cross-validation
#define a loss function (average prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
#compute the average number of wrong predictions in the (training) data
#K-fold cross-validation
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m, K = nrow(alc))
#average number of wrong predictions in the cross validation
cv$delta[1]## [1] 0.2539267# 10-fold cross-validation
cost <- function(r, pi = 0) mean(abs(r-pi) > 0.5)
alc.glm <- glm(high_use ~ famrel+goout, binomial, data = alc)
(cv.err <- cv.glm(alc, alc.glm, cost, K = nrow(alc))$delta)## [1] 0.2460733 0.2460733(cv.11.err <- cv.glm(alc, alc.glm, cost, K = 10)$delta)## [1] 0.2696335 0.2619720## [1] 0.2643979 0.2548998Here a made a cross-validation from the DataCamp + made myself a 10-fold cross-validation. My model has a little bit higher prediction error using 10-fold cross-validation) compared to the model introduced in DataCamp (which had about 0.26 error). So it is not better.
#cross-validation with 6 predictors
cost <- function(r, pi = 0) mean(abs(r-pi) > 0.5)
alc.glm <- glm(high_use ~ famrel+goout+sex+absences+freetime+paid, binomial, data = alc)
(cv.err <- cv.glm(alc, alc.glm, cost, K = nrow(alc))$delta)## [1] 0.2225131 0.2205532(cv.11.err <- cv.glm(alc, alc.glm, cost, K = 10)$delta)## [1] 0.2303665 0.2265289## [1] 0.2198953 0.2154409#cross-validation with 4 predictors: sex, absences, freetime, paid
cost <- function(r, pi = 0) mean(abs(r-pi) > 0.5)
alc.glm <- glm(high_use ~ sex+absences+freetime+paid, binomial, data = alc)
(cv.err <- cv.glm(alc, alc.glm, cost, K = nrow(alc))$delta)## [1] 0.2591623 0.2594707(cv.11.err <- cv.glm(alc, alc.glm, cost, K = 10)$delta)## [1] 0.2617801 0.2617801## [1] 0.2670157 0.2659809#cross-validation with 4 predictors: famrel, goout, absences, freetime
cost <- function(r, pi = 0) mean(abs(r-pi) > 0.5)
alc.glm <- glm(high_use ~ famrel+goout+absences+freetime, binomial, data = alc)
(cv.err <- cv.glm(alc, alc.glm, cost, K = nrow(alc))$delta)## [1] 0.2486911 0.2479510(cv.11.err <- cv.glm(alc, alc.glm, cost, K = 10)$delta)## [1] 0.2565445 0.2528919## [1] 0.2460733 0.2415641#cross-validation with 2 predictors: famrel, absences
cost <- function(r, pi = 0) mean(abs(r-pi) > 0.5)
alc.glm <- glm(high_use ~ famrel+absences, binomial, data = alc)
(cv.err <- cv.glm(alc, alc.glm, cost, K = nrow(alc))$delta)## [1] 0.2931937 0.2938516(cv.11.err <- cv.glm(alc, alc.glm, cost, K = 10)$delta)## [1] 0.2879581 0.2931320## [1] 0.2931937 0.2966476Here I Perform cross-validation to compare the performance of different logistic regression models. I first test model with such predictors as famrel, goout, sex, absences, freetime, paid. It is guite a good model since the errors are lower than before and the training error is less than the test error. Now let’s try to decrease the number of predictors. I have tried the model with 4 predictors: sex, absences, freetime, paid. Here again errors are higher and test erroes are higher. I also tried model with such 4 predictors: famrel, goout, absences, freetime. Here errors are higher than in model with 6 predictors but lower than in the previous model. Also testing errors in 10 validation are less than in training, so the model is quite good. In the end I am trying the model with 2 predictors:family relations and absences. Errors are the highest but the testing ones are less than training.
Conclusion:The higher number of the predictors result in the lower errors of the model since the more factors we have the much our model explains. ***
Exercise 4
## Data overview
library(MASS)##
## Attaching package: 'MASS'## The following object is masked from 'package:dplyr':
##
## selectdata("Boston")
str(Boston)## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...summary(Boston)## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00#variables distribution
p <- pairs(Boston)
p## NULL#calculate the correlation matrix and round it
cor_matrix<-cor(Boston)
#print the correlation matrix
cor_matrix## crim zn indus chas nox
## crim 1.00000000 -0.20046922 0.40658341 -0.055891582 0.42097171
## zn -0.20046922 1.00000000 -0.53382819 -0.042696719 -0.51660371
## indus 0.40658341 -0.53382819 1.00000000 0.062938027 0.76365145
## chas -0.05589158 -0.04269672 0.06293803 1.000000000 0.09120281
## nox 0.42097171 -0.51660371 0.76365145 0.091202807 1.00000000
## rm -0.21924670 0.31199059 -0.39167585 0.091251225 -0.30218819
## age 0.35273425 -0.56953734 0.64477851 0.086517774 0.73147010
## dis -0.37967009 0.66440822 -0.70802699 -0.099175780 -0.76923011
## rad 0.62550515 -0.31194783 0.59512927 -0.007368241 0.61144056
## tax 0.58276431 -0.31456332 0.72076018 -0.035586518 0.66802320
## ptratio 0.28994558 -0.39167855 0.38324756 -0.121515174 0.18893268
## black -0.38506394 0.17552032 -0.35697654 0.048788485 -0.38005064
## lstat 0.45562148 -0.41299457 0.60379972 -0.053929298 0.59087892
## medv -0.38830461 0.36044534 -0.48372516 0.175260177 -0.42732077
## rm age dis rad tax
## crim -0.21924670 0.35273425 -0.37967009 0.625505145 0.58276431
## zn 0.31199059 -0.56953734 0.66440822 -0.311947826 -0.31456332
## indus -0.39167585 0.64477851 -0.70802699 0.595129275 0.72076018
## chas 0.09125123 0.08651777 -0.09917578 -0.007368241 -0.03558652
## nox -0.30218819 0.73147010 -0.76923011 0.611440563 0.66802320
## rm 1.00000000 -0.24026493 0.20524621 -0.209846668 -0.29204783
## age -0.24026493 1.00000000 -0.74788054 0.456022452 0.50645559
## dis 0.20524621 -0.74788054 1.00000000 -0.494587930 -0.53443158
## rad -0.20984667 0.45602245 -0.49458793 1.000000000 0.91022819
## tax -0.29204783 0.50645559 -0.53443158 0.910228189 1.00000000
## ptratio -0.35550149 0.26151501 -0.23247054 0.464741179 0.46085304
## black 0.12806864 -0.27353398 0.29151167 -0.444412816 -0.44180801
## lstat -0.61380827 0.60233853 -0.49699583 0.488676335 0.54399341
## medv 0.69535995 -0.37695457 0.24992873 -0.381626231 -0.46853593
## ptratio black lstat medv
## crim 0.2899456 -0.38506394 0.4556215 -0.3883046
## zn -0.3916785 0.17552032 -0.4129946 0.3604453
## indus 0.3832476 -0.35697654 0.6037997 -0.4837252
## chas -0.1215152 0.04878848 -0.0539293 0.1752602
## nox 0.1889327 -0.38005064 0.5908789 -0.4273208
## rm -0.3555015 0.12806864 -0.6138083 0.6953599
## age 0.2615150 -0.27353398 0.6023385 -0.3769546
## dis -0.2324705 0.29151167 -0.4969958 0.2499287
## rad 0.4647412 -0.44441282 0.4886763 -0.3816262
## tax 0.4608530 -0.44180801 0.5439934 -0.4685359
## ptratio 1.0000000 -0.17738330 0.3740443 -0.5077867
## black -0.1773833 1.00000000 -0.3660869 0.3334608
## lstat 0.3740443 -0.36608690 1.0000000 -0.7376627
## medv -0.5077867 0.33346082 -0.7376627 1.0000000#visualize the correlation matrix
library(corrplot)## corrplot 0.84 loadedlibrary(tidyverse)## ── Attaching packages ───────────────────────────────── tidyverse 1.2.1 ──## ✔ tibble 2.1.3 ✔ purrr 0.3.3
## ✔ readr 1.3.1 ✔ stringr 1.4.0
## ✔ tibble 2.1.3 ✔ forcats 0.4.0## ── Conflicts ──────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag() masks stats::lag()
## ✖ MASS::select() masks dplyr::select()corrplot(cor_matrix, method="circle")
corrplot(cor_matrix, method="circle", type = "upper",cl.pos = "b", tl.pos = "d",tl.cex = 0.6)%>%round()
## crim zn indus chas nox rm age dis rad tax ptratio black lstat medv
## crim 1 0 0 0 0 0 0 0 1 1 0 0 0 0
## zn 0 1 -1 0 -1 0 -1 1 0 0 0 0 0 0
## indus 0 -1 1 0 1 0 1 -1 1 1 0 0 1 0
## chas 0 0 0 1 0 0 0 0 0 0 0 0 0 0
## nox 0 -1 1 0 1 0 1 -1 1 1 0 0 1 0
## rm 0 0 0 0 0 1 0 0 0 0 0 0 -1 1
## age 0 -1 1 0 1 0 1 -1 0 1 0 0 1 0
## dis 0 1 -1 0 -1 0 -1 1 0 -1 0 0 0 0
## rad 1 0 1 0 1 0 0 0 1 1 0 0 0 0
## tax 1 0 1 0 1 0 1 -1 1 1 0 0 1 0
## ptratio 0 0 0 0 0 0 0 0 0 0 1 0 0 -1
## black 0 0 0 0 0 0 0 0 0 0 0 1 0 0
## lstat 0 0 1 0 1 -1 1 0 0 1 0 0 1 -1
## medv 0 0 0 0 0 1 0 0 0 0 -1 0 -1 1This data has 506 obs. of 14 variables which contain information about different indicators of Boston city as per capita crime rate, index of accessibility to radial highways and etc. In the first plot we see the distibution of variables. It looks like only rm has a normal distribution while crime, chas, lstat are shifted to the left, age to the right and etc. In the second plot positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients. For example, the is strong positive correlation between such variables as crime and rad and tax, between indus and nox, age and tax. Negative correlations between dis and nox,age and dis, medv and lstat.
=======This data has 506 obs. of 14 variables which contain information about different indicators of Boston city as per capita crime rate, index of accessibility to radial highways and etc. In the fist plot we seen the distibution of variables. It looks like only rm has a normal distribution while crime, chas, lstat are shifted to the left, age to the right and etc. In the second plot positive correlations are displayed in blue and negative correlations in red color. Color intensity and the size of the circle are proportional to the correlation coefficients. For example, the is strong positive correlation between such variables as crime and rad and tax, between indus and nox, age and tax. Negative correlations between dis and nox,age and dis, medv and lstat.
>>>>>>> 77475a1150362e7e1077341fd0cd3c34d7f8d6cc## Scaling and factor variable
#center and standardize variables
boston_scaled <- scale (Boston)
#summaries of the scaled variables
summary(boston_scaled) ## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865#class of the boston_scaled object
class(boston_scaled)## [1] "matrix"#change the object to data frame
boston_scaled <- as.data.frame(boston_scaled)
#create a quantile vector of crim and print it
bins <- quantile(boston_scaled$crim)
bins## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610#create a categorical variable 'crime'
vector <- c("low","med_low","med_high","high")
crime <- cut(boston_scaled$crim, breaks = bins, include.lowest = TRUE, label=vector)
#look at the table of the new factor crime
table(crime)## crime
## low med_low med_high high
## 127 126 126 127# remove original crim from the dataset
boston_scaled <- dplyr::select(boston_scaled, -crim)
#add the new categorical value to scaled data
boston_scaled <- data.frame(boston_scaled, crime)
## Train and test sets
#number of rows in the Boston dataset
n <- nrow(boston_scaled)
#choose randomly 80% of the rows
ind <- sample(n, size = n * 0.8)
#create train set
train <- boston_scaled[ind,]
#create test set
test <- boston_scaled[-ind,]
#save the correct classes from test data
correct_classes <- c(test$crime)
#remove the crime variable from test data
test <- dplyr::select(test, -crime)
str(train)## 'data.frame': 404 obs. of 14 variables:
## $ zn : num -0.487 -0.487 -0.487 -0.487 -0.487 ...
<<<<<<< HEAD
## $ indus : num -1.033 1.015 -0.164 2.42 -0.72 ...
## $ chas : num -0.272 -0.272 -0.272 -0.272 3.665 ...
## $ nox : num -0.3857 0.3651 -0.0664 0.4686 -0.4116 ...
## $ rm : num 0.187 0.53 -0.508 -0.429 0.628 ...
## $ age : num 0.5515 1.1164 0.6972 0.5302 -0.0737 ...
## $ dis : num -0.546 -1.077 -0.633 -0.8 -0.068 ...
## $ rad : num -0.522 1.66 -0.408 -0.637 -0.178 ...
## $ tax : num -0.666 1.529 0.141 1.796 -0.601 ...
## $ ptratio: num -0.857 0.806 -0.303 0.76 -0.488 ...
## $ black : num 0.4253 -3.5229 -0.1289 0.4406 0.0386 ...
## $ lstat : num -0.506 1.1997 0.4351 0.0976 -0.6446 ...
## $ medv : num 0.116 -0.58 -0.46 -0.265 0.703 ...
## $ crime : Factor w/ 4 levels "low","med_low",..: 2 4 2 2 3 4 2 1 2 4 ...str(test)## 'data.frame': 102 obs. of 13 variables:
## $ zn : num 0.2845 0.0487 0.0487 -0.4872 -0.4872 ...
## $ indus : num -1.287 -0.476 -0.476 -0.437 -0.437 ...
## $ chas : num -0.272 -0.272 -0.272 -0.272 -0.272 ...
## $ nox : num -0.144 -0.265 -0.265 -0.144 -0.144 ...
## $ rm : num 0.413 -0.93 0.131 -1.179 -0.455 ...
## $ age : num -0.12 1.116 0.914 -1.136 0.733 ...
## $ dis : num 0.140075 1.086122 1.21178 0.000692 0.103175 ...
## $ rad : num -0.982 -0.522 -0.522 -0.637 -0.637 ...
## $ tax : num -0.666 -0.577 -0.577 -0.601 -0.601 ...
## $ ptratio: num -1.46 -1.5 -1.5 1.18 1.18 ...
## $ black : num 0.441 0.328 0.393 -0.741 0.393 ...
## $ lstat : num -1.074 2.419 1.092 -0.135 0.165 ...
## $ medv : num 0.16 -0.656 -0.819 -0.254 -0.319 ...str(test)## 'data.frame': 102 obs. of 13 variables:
## $ zn : num -0.4872 -0.4872 0.0487 0.0487 0.0487 ...
## $ indus : num -0.593 -0.593 -0.476 -0.476 -0.476 ...
## $ chas : num -0.272 -0.272 -0.272 -0.272 -0.272 ...
## $ nox : num -0.74 -0.74 -0.265 -0.265 -0.265 ...
## $ rm : num 0.194 1.281 -0.16 -0.93 -0.563 ...
## $ age : num 0.367 -0.266 0.978 1.116 -1.051 ...
## $ dis : num 0.557 0.557 1.024 1.086 0.786 ...
## $ rad : num -0.867 -0.867 -0.522 -0.522 -0.522 ...
## $ tax : num -0.986 -0.986 -0.577 -0.577 -0.577 ...
## $ ptratio: num -0.303 -0.303 -1.504 -1.504 -1.504 ...
## $ black : num 0.441 0.396 0.441 0.328 0.371 ...
## $ lstat : num -0.492 -1.208 0.91 2.419 0.428 ...
## $ medv : num -0.1014 1.3229 0.4966 -0.6559 -0.0906 ...Here we scale the data which is an operation when we subtract the column means from the corresponding columns and divide the difference with standard deviation. It helps us to have normal distribution of variables later used in clastering. When we create a factor variable’crim’ and use the quantiles as the break points to the variable.Later we divide the dataset to train and test sets, so that 80% of the data belongs to the train set.
# Linear discriminant analysis
lda.fit <- lda(crime ~ ., data = train)
#print the lda.fit object
lda.fit## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
<<<<<<< HEAD
## 0.2500000 0.2351485 0.2599010 0.2549505
##
## Group means:
## zn indus chas nox rm
## low 0.91794122 -0.8964476 -0.155385496 -0.8708570 0.4612307
## med_low -0.07742503 -0.3407435 -0.106556426 -0.5896536 -0.1173697
## med_high -0.38025152 0.1117786 0.177625245 0.3633568 0.1183384
## high -0.48724019 1.0170891 -0.004759149 1.0366145 -0.3987911
## age dis rad tax ptratio
## low -0.8740888 0.8265347 -0.6953230 -0.7578877 -0.41118220
## med_low -0.3824463 0.4012400 -0.5357824 -0.4911322 -0.07281424
## med_high 0.3565856 -0.3583810 -0.3890435 -0.3109617 -0.31379225
## high 0.8228049 -0.8454307 1.6384176 1.5142626 0.78111358
## black lstat medv
## low 0.3853158 -0.78525002 0.55762985
## med_low 0.3141281 -0.20235972 0.03385894
## med_high 0.1143596 0.01385463 0.16646579
## high -0.7720726 0.92007817 -0.69416009
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.0846507053 0.76548460 -0.99101074
## indus -0.0008345807 -0.19960977 0.31553665
## chas -0.0641603165 -0.10027899 0.07210062
## nox 0.3675570896 -0.66628970 -1.34551901
## rm -0.0958845586 -0.11424536 -0.21780873
## age 0.3117103860 -0.30660388 -0.03821072
## dis -0.0517525721 -0.24641974 0.34604917
## rad 2.9517261386 0.94314128 -0.03558019
## tax 0.0541839210 -0.08115445 0.52756593
## ptratio 0.1025626309 0.11170313 -0.32149396
## black -0.1191068792 -0.03112145 0.07744718
## lstat 0.2400950484 -0.26590388 0.32710883
## medv 0.1845112388 -0.31357167 -0.13482788
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9489 0.0385 0.0126#the function for lda biplot arrows
lda.arrows <- function(x, myscale = 1, arrow_heads = 0.1, color = "red", tex = 0.75, choices = c(1,2)){
heads <- coef(x)
arrows(x0 = 0, y0 = 0,
x1 = myscale * heads[,choices[1]],
y1 = myscale * heads[,choices[2]], col=color, length = arrow_heads)
text(myscale * heads[,choices], labels = row.names(heads),
cex = tex, col=color, pos=3)
}
#target classes as numeric
classes <- as.numeric(train$crime)
#plot the lda results
plot(lda.fit, dimen = 2,col = classes,pch = classes)
lda.arrows(lda.fit, myscale = 2)
We have 4 clasters, so train data was devided in 25% 4 times with crime as target variable. In the plot we see that three clasters are overlapping while the 4rd one in quite far for them. We aslo see that such variables as zn, nox, rad, ptratio have a big impact on the model
# Predictors
#create train set
train <- boston_scaled[ind,]
#create test set
test <- boston_scaled[-ind,]
#save the correct classes from test data
correct_classes <- c(test$crime)
#remove the crime variable from test data
test <- dplyr::select(test, -crime)
#predict classes with test data
lda.pred <- predict(lda.fit, newdata = test)
# cross tabulate the results
table(correct = correct_classes, predicted = lda.pred$class)## predicted
## correct low med_low med_high high
<<<<<<< HEAD
## 1 16 10 0 0
## 2 6 14 11 0
## 3 0 2 19 0
## 4 0 0 0 24#target classes as numeric
classes <- as.numeric(correct_classes)
#plot the lda results
plot(lda.fit, dimen = 2,col = classes,pch = classes)
lda.arrows(lda.fit, myscale = 2)
In the table we see the relation between the correct classes and the predicted ones. The classifier did not predict the crime rates correctly since predicted numbers are higher than correct. I also tried to vizualize the LDA results for crime in test data and we see that everything is in a mess.
## distances
library(MASS)
data('Boston')
#scale the data
boston_scaled <- scale (Boston)
#euclidean distance matrix
dist_eu <- (boston_scaled)
#look at the summary of the distances
summary(dist_eu)## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865#manhattan distance matrix
dist_man <- dist(boston_scaled, method = "manhattan")
#look at the summary of the distances
summary(dist_man)## Min. 1st Qu. Median Mean 3rd Qu. Max.
## 0.2662 8.4832 12.6090 13.5488 17.7568 48.8618# k-means clustering
km <-kmeans(boston_scaled, centers = 1)
km <-kmeans(boston_scaled, centers = 3)
km <-kmeans(boston_scaled, centers = 2)
# plot the Boston dataset with clusters
pairs(boston_scaled, col=km$cluster)
#determine K
set.seed(123)
library(ggplot2)
#determine the number of clusters
k_max <- 10
#calculate the total within sum of squares
twcss <- sapply(1:k_max, function(k){kmeans(boston_scaled, k)$tot.withinss})
#visualize the results
qplot(x = 1:k_max, y = twcss, geom = 'line')
#k-means clustering
km <-kmeans(boston_scaled, centers = 2)Based on the calculated distance meajure the k-classtering was maded. On the graph we see a significant change in point 2 - so the optimal number of clastters is 2. On the plot we see the vizualization for 2 clasters - which part of data where belongs.
## 3D plot
model_predictors <- dplyr::select(train, -crime)
# check the dimensions
dim(model_predictors)## [1] 404 13dim(lda.fit$scaling)## [1] 13 3# matrix multiplication
matrix_product <- as.matrix(model_predictors) %*% lda.fit$scaling
matrix_product <- as.data.frame(matrix_product)
library(plotly)##
## Attaching package: 'plotly'## The following object is masked from 'package:MASS':
##
## select## The following object is masked from 'package:ggplot2':
##
## last_plot## The following object is masked from 'package:stats':
##
## filter## The following object is masked from 'package:graphics':
##
## layoutplot_ly(x = matrix_product$LD1, y = matrix_product$LD2, z = matrix_product$LD3, type= 'scatter3d', mode='markers')